Source-free ModMax theories[1]} of nonlinear electrodynamics (NLE) in the four dimensional Minkowski spacetime vacuum
are the only possible continuous deformations of source-free Maxwell linear electrodynamics (MLE) in the same vacuum,
which preserve all the same continuous Poincaré and conformal spacetime symmetries and duality invariances of MLE.
These ModMax theories are labelled by a single real and positive parameter \(\gamma \ge 0\) , such that ordinary Maxwell theory
is recovered for \(\gamma =0\) . Null field configurations play a central role for MLE. They include not only monochromatic travelling plane waves,
for instance, but null electromagnetic knots as well with their topologically nontrivial structures, among which hopfion-Rañada knots constitute
a class of their own. A priori any such null configuration remains, without deformation, an exact solution for any Poincaré invariant NLE theory,
provided however that spacetime conformal invariance is no longer in place. By lack of an intrinsic physical scale, in the case of a spacetime Poincaré
and conformal invariant NLE, null field configurations are ill-defined within the Lagrangian formulation, because of a lack
of analyticity in that case of the Lagrangian density as a function of the electromagnetic Lorentz invariants. Nonetheless because of their topologically
nontrivial structure null knots are expected to remain robust against whatever continuous nonlinear deformations of MLE, which they are indeed
without deformation when spacetime conformal invariance is not enforced.
